The relation between resonant frequencies and torsional stiffness of long bones in vitro. Validation of a simple beam model

Abstract

The results of vibration analysis experiments and impact torsion tests performed on excised animal long bones were used to validate a simple beam model for the prediction of torsional stiffness from resonant frequencies. Resonant frequency data on two mutually perpendicular bending vibration modes of 142 excised long bones were evaluated. Torsional stiffness of the same bones had been determined by an impact torsion test. Using a simple beam model, a theoretical relation between resonant frequencies and torsional stiffness was derived. If total bone mass and bone length are known, the formula thus derived allows one to calculate torsional stiffness from resonant frequencies. Linear regression analysis shows a strong correlation between the measured and the calculated torsional stiffness for sheep femora ( r 2 = 0.63, n = 24), dog femora ( r 2 = 0.94, n = 34), dog tibiae ( r 2 = 0.79, n = 18) and monkey radii ( r 2 = 0.77, n = 66). It was found that this linear relation was valid not within one bone type alone. Linear regression analysis on the combined data of all bones demonstrated that all bones obeyed the same global linear relation between measured and the calculated torsional stiffness ( r 2 = 0.98, n = 142). This implies that one and the same beam model is valid for the different bone types investigated. The calculation of stiffness from resonant frequencies, however, requires total bone mass, m, and length to be known. In view of in vivo applications, the feasibility of using total bone mineral content (TBMC) as a measure for m was investigated. A good linear correlation was found between total bone mass and total bone mineral content. The correlation between torsional stiffness using TBMC data and measured torsional stiffness was still highly satisfying. Finally, a good linear correlation was observed between the properties of left and right bone from a single specimen. These results demonstrate a simple relation between resonant frequencies and bone mechanical properties in vitro. Under the condition that in vivo effects (muscles, skin, joints, etc.) on the vibrational behaviour can be accounted for properly, the results are promising as to the feasibility of the resonant frequency technique for fracture healing monitoring and osteoporosis evaluation.